Found problems: 161
2023 Harvard-MIT Mathematics Tournament, 15
Let $A$ and $B$ be points in space such that $AB=1.$ Let $\mathcal{R}$ be the region of points $P$ for which $AP \le 1$ and $BP \le 1.$ Compute the largest possible side length of a cube contained in $\mathcal{R}.$
2024 HMNT, 2
Compute the smallest integer $n > 72$ that has the same set of prime divisors as $72.$
2024 LMT Fall, 10
David starts at the point $A$ and goes up and right along the grid lines to point $B$. At each of the points $C$, $D$, and $E$ there is a bully. Find the number of paths David can take which make him encounter exactly one bully.
[asy]
size(150);
draw((0,0)--(4,0)--(4,3)--(0,3)--cycle);
draw((0,1)--(4,1));
draw((0,2)--(4,2));
draw((1,0)--(1,3));
draw((2,0)--(2,3));
draw((3,0)--(3,3));
dot((0,0)); label("A", (0,0), W);
dot((4,3)); label("B", (4,3), E);
dot((1,1.5)); label("C", (1,1.5), W);
dot((2,0.5)); label("D", (2,0.5), W);
dot((2.5,2)); label("E", (2.5,2), N);
[/asy]
2024 Harvard-MIT Mathematics Tournament, 8
Three points, $A, B,$ and $C,$ are selected independently and uniformly at random from the interior of a unit square. Compute the expected value of $\angle{ABC}.$
2024 Harvard-MIT Mathematics Tournament, 12
Compute the number of quadruples $(a,b,c,d)$ of positive integers satisfying $$12a+21b+28c+84d=2024.$$
2024 LMT Fall, 22
Find the number of real numbers $0 \leq \alpha < 50$ such that $\alpha^2 + 2\{\alpha\}$ is an integer. (Here $\{\alpha\}$ denotes the fractional part of $\alpha$.)
2024 HMNT, 6
The vertices of a cube are labeled with the integers $1$ through $8,$ with each used exactly once. Let $s$ be the maximum sum of the labels of two edge-adjacent vertices. Compute the minimum possible value of $s$ over all such labelings.
2023 BMT, 27
Let $\omega$ be a circle with positive integer radius $r$. Suppose that it is possible to draw isosceles triangle with integer side lengths inscribed in $\omega$. Compute the number of possible values of $r$ where $1 \le r \le 2023^2$.
Submit your answer as a positive integer $E$. If the correct answer is $A$, your score for this question will be $\max \left( 0, 25\left(3 - 2 \max \left( \frac{A}{E} , \frac{E}{A}\right)\right)\right)$, rounded to the nearest integer.
2025 Harvard-MIT Mathematics Tournament, 19
A subset $S$ of $\{1, 2, 3, \ldots, 2025\}$ is called [i]balanced[/i] if for all elements $a$ and $b$ both in $S,$ there exists an element $c$ in $S$ such that $2025$ divides $a+b-2c.$ Compute the number of [i]nonempty[/i] balanced sets.
2023 Harvard-MIT Mathematics Tournament, 29
Let $P_1(x), P_2(x), \ldots, P_k(x)$ be monic polynomials of degree $13$ with integer coefficients. Suppose there are pairwise distinct positive integers $n_1, n_2, \ldots, n_k$ for which, for all positive integers $i$ and $j$ less than or equal to $k,$ the statement "$n_i$ divides $P_j(m)$ for every integer $m$" holds if and only if $i=j.$ Compute the largest possible value of $k.$
2024 Harvard-MIT Mathematics Tournament, 7
Positive integers $a, b,$ and $c$ have the property that $a^b, b^c,$ and $c^a$ end in $4, 2,$ and $9,$ respectively. Compute the minimum possible value of $a+b+c.$
2024 LMT Fall, 27
Find all positive integer pairs $(a,b)$ that satisfy the equation$$a^2b+ab^2+73=8ab+9a+9b.$$
2024 Harvard-MIT Mathematics Tournament, 23
Let $\ell$ and $m$ be two non-coplanar lines in space, and let $P_1$ be a point on $\ell.$ Let $P_2$ be the point on $,m$ closest to $P_1,$ $P_3$ be the point on $\ell$ closest to $P_3,$ $P_4$ be the point on $m$ closest to $P_3,$ and $P_5$ be the point on $\ell$ closest to $P_4.$ Given that $P_1P_2=5, P_2P_3=3,$ and $P_3P_4=2,$ compute $P_4P_5.$
2025 Harvard-MIT Mathematics Tournament, 15
Right triangle $\triangle{DEF}$ with $\angle{D}=90^\circ$ and $\angle{F}=30^\circ$ is inscribed in equilateral triangle $\triangle{ABC}$ such that $D, E,$ and $F$ lie on segments $\overline{BC}, \overline{CA},$ and $\overline{AB},$ respectively. Given that $BD=7$ and $DC=4,$ compute $DE.$
2023 Harvard-MIT Mathematics Tournament, 16
The graph of the equation $x+y=\lfloor x^2+y^2 \rfloor$ consists of several line segments. Compute the sum of their lengths.
2023 Harvard-MIT Mathematics Tournament, 24
Let $AXBY$ be a cyclic quadrilateral, and let line $AB$ and line $XY$ intersect at $C.$ Suppose $AX \cdot AY = 6, BX \cdot BY=5,$ and $CX \cdot CY=4.$ Compute $AB^2.$
2024 HMNT, 15
Compute the sum of the three smallest positive integers $n$ for which $$\frac{1+2+3+\cdots+(2024n-1)+2024n}{1+2+3+\cdots+(4n-1)+4n}$$ is an integer.
2024 HMNT, 29
Let $ABC$ be a triangle such that $AB = 3, AC = 4,$ and $\angle{BAC} = 75^\circ.$ Square $BCDE$ is constructed outside triangle $ABC.$ Compute $AD^2 +AE^2.$
2024 Harvard-MIT Mathematics Tournament, 9
Compute the sum of all positive integers $n$ such that $n^2-3000$ is a perfect square.
2024 HMNT, 31
Positive integers $a, b,$ and $c$ have the property that $\text{lcm}(a,b), \text{lcm}(b,c),$ and $\text{lcm}(c,a)$ end in $4, 6,$ and $7,$ respectively, when written in base $10.$ Compute the minimum possible value of $a + b + c.$
2025 Harvard-MIT Mathematics Tournament, 2
Compute $$\frac{20+\frac{1}{25-\frac{1}{20}}}{25+\frac{1}{20-\frac{1}{25}}}.$$
2023 Harvard-MIT Mathematics Tournament, 9
One hundred points labeled $1$ to $100$ are arranged in a $10\times 10$ grid such that adjacent points are one unit apart. The labels are increasing left to right, top to bottom (so the first row has labels $1$ to $10,$ the second row has labels $11$ to $20,$ and so on).
Convex polygon $\mathcal{P}$ has the property that every point with a label divisible by $7$ is either on the boundary or in the interior of $\mathcal{P}.$ Compute the smallest possible area of $\mathcal{P}.$
2025 Harvard-MIT Mathematics Tournament, 12
Holden has a collection of polygons. He writes down a list containing the measure of each interior angle of each of his polygons. He writes down the list $30^\circ, 50^\circ, 60^\circ, 70^\circ, 90^\circ, 100^\circ, 120^\circ, 160^\circ,$ and $x^\circ,$ in some order. Compute $x.$
2024 HMNT, 33
A grid is called [i]groovy[/i] if each cell of the grid is labeled with the smallest positive integer that does not appear below it in the same column or to the left of it in the same row. Compute the sum of the entries of a groovy $14 \times 14$ grid whose bottom left entry is $1.$
2023 Harvard-MIT Mathematics Tournament, 32
Let $ABC$ be a triangle with $\angle{BAC}>90^\circ.$ Let $D$ be the foot of the perpendicular from $A$ to side $BC.$ Let $M$ and $N$ be the midpoints of segments $BC$ and $BD,$ respectively. Suppose that $AC=2, \angle{BAN}=\angle{MAC},$ and $AB \cdot BC = AM.$ Compute the distance from $B$ to line $AM.$